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Mirrors > Home > MPE Home > Th. List > fmptapd | Structured version Visualization version GIF version |
Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.) |
Ref | Expression |
---|---|
fmptapd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
fmptapd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
fmptapd.s | ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) |
fmptapd.c | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) |
Ref | Expression |
---|---|
fmptapd | ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptapd.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) | |
2 | fmptapd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | fmptapd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | 1, 2, 3 | fmptsnd 7116 | . . 3 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶)) |
5 | 4 | uneq2d 4124 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))) |
6 | mptun 6648 | . . 3 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)) | |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))) |
8 | fmptapd.s | . . 3 ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) | |
9 | 8 | mpteq1d 5201 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
10 | 5, 7, 9 | 3eqtr2d 2779 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∪ cun 3909 {csn 4587 ⟨cop 4593 ↦ cmpt 5189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-opab 5169 df-mpt 5190 |
This theorem is referenced by: fmptpr 7119 poimirlem3 36127 poimirlem4 36128 poimirlem16 36140 poimirlem17 36141 poimirlem19 36143 poimirlem20 36144 |
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